# The Conjugate Main Eigenvalues Of Vertex Deleted Subgraphs of a Strongly Regular Graph

### New Zealand Journal of Mathematics

Vol. 40, (2010), Pages 67-74

Mirko LepoviÄ‡

Tihomira VuksanoviÄ‡a 32,

34000, Kragujevac,

Serbia

Abstract Let G be a simple graph of order n. Let $c = a + b\sqrt{m}$ and $\overline c = a - b\sqrt{m}$, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that Ac = [cij] is the conjugate adjacency matrix of the graph G if cij = c for any two adjacent vertices i and j, $c_{ij} = \overline c$ for any two nonadjacent vertices i and j, and cij = 0 if i = j. Let $(A^c)^k = [c_{ij}^{(k)}]$ for any nonnegative integer k. Further, let G be a strongly regular graph of degree $r\ge 1$, understanding that G is not the complete graph. Then $c_{ij}^{(2)} = \tau^c$ for any two adjacent vertices i and j and $c_{ij}^{(2)}=\theta^c$ for any two distinct nonadjacent vertices i and j, where τc and θc are two fixed real numbers. Let $r^c = rc + ((n-1) -r)\,\overline c$. We demonstrate that

$\mu_{1,2}^c = \frac{\tau^c - \theta^c + \big(c-\overline c\big)r^c \pm \sqrt{\big(\tau^c - \theta^c - (c-\overline c)\,r^c\big)^2 + 4(c-\overline c) \big(\overline c\,\tau^c - c\,\theta^c\big)}}{2\big(c-\overline c\big)}$

where $\mu_1^c$ and $\mu_2^c$ are the conjugate main eigenvalues of its vertex deleted subgraphs $G_i = G \smallsetminus i$ for i = 1,2,...,n.

Keywords strongly regular graph, conjugate adjacency matrix, conjugate main eigenvalue.

Classification (MSC2000) 05C50